Q.In a perfectly symmetric distribution, Karl Pearson's coefficient of skewness is:

In a symmetric distribution the mean and mode are identical, so the numerator of Karl Pearson's formula is zero. …

In a perfectly symmetric distribution, Mean = Median = Mode (Section 1 of this chapter). Karl Pearson's coefficient is:

Since Mean and Mode are equal in a symmetric distribution, the numerator $(\text{Mean}-\text{Mode})=0$, and therefore $Sk_P=0$ regardless of the value of the standard deviation.

Why the other options are wrong:

• (a) Positive would require Mean > Mode, which contradicts symmetry.
• (b) Negative would require Mean < Mode, which also contradicts symmetry. …

Do not confuse the theoretical range of the coefficient (−3 to +3) with its value for a symmetric distribution, which …